where \(T\)  is the temperature and \(k_B\)  is Boltzmann's constant.
The ensemble average of the extracted work is defined as
\(\langle W \rangle = \sum _{k = -N/2}^{N/2} W(k) \Pr (k)\) , with \(k\)  distributed binomially (shifted to \(\langle k \rangle =0\) ), \(\Pr (k) = \left(\frac{1}{2}\right)^N \binom{N}{\frac{N}{2}+k}\) , since particles are non-interacting and their spatial distributions are uniform. We note that for the engine operating on an ideal gas and \(N\gg 1\) , a useful approximation for the mean work can be obtained by noting that \(P(k)\)  is appreciably large when \(|k/N|\ll 1\) . As a result one can expand \(W(k)\)  in powers of \(k/N\)  [1]},
\(&W(k) = N k_B T \left(2\left(\frac{k}{N}\right)^2 + O \left(\left(\frac{k}{N}\right)^4\right)\right), \\&\lim _{N\rightarrow \infty }\langle W_{\mathrm {ideal}} \rangle = \frac{2 k_B T}{N} \langle k^2 \rangle = \frac{1}{2} k_B T, \) 
